Unconventional magnetoelectric conductivity and electrochemical response from dipole-like sources of Berry curvature

TL;DR

This work analyzes unconventional magnetoelectric and electrochemical responses in three-dimensional topological semimetals with zero Chern number: vortex nodal-ring (BC dipole) and three-band Hopf semimetals (BC dipole nodes), plus PT-symmetric nodal rings for comparison. Using exact semiclassical Boltzmann theory, it computes the longitudinal magnetoconductivity $σ_{zz}$ under collinear electric and magnetic fields along $\hat{z}$, incorporating Berry-curvature and orbital magnetic moment corrections, and demonstrates linear-in-$|B|$ as well as quadratic-in-$|B|$ contributions arising from BC anisotropy. It then develops nonlinear electrochemical response (ECR) within the relaxation-time framework, expressed via third-rank tensors and a Berry-curvature dipole tensor $D^{s}_{ab}$, revealing nonzero ECR components in tilted configurations and relating them to BC/dipole textures. The results reveal a strong link between BC-dipole topology and anisotropic transport, suggesting that BC dipoles can yield signatures akin to monopole cases, and propose future exploration of noncollinear field configurations to probe Hall- and planar-Hall-type responses in these systems.

Abstract

We compute longitudinal magnetoelectric conductivity ($σ_{zz}$) and nonlinear electrochemical response (ECR), applying the semiclassical Boltzmann formalism, for three-dimensional nodal-ring semimetals (vortex nodal-rings and $\mathcal P \mathcal T$-symmetric nodal-rings) and three-band Hopf semimetals. While the nodal-curves of the former are taken to lie along the $k_z = 0$-plane, the nodal points of the latter harbour dipoles in their Berry-curvature (BC) profile, with the dipole's axis aligned along the $k_z$-axis. All these systems are topological and are unified on the aspect that their bands possess a vanishing Chern number. The linear response, $σ_{zz}$, is obtained from an exact solution when the systems are subjected to collinear electric and magnetic fields applied along the anisotropy axis, viz. $\boldsymbol{\hat z}$. The nonlinear part involves third-rank tensors representing second-order response coefficients, relating the electrical current to the combined effects of the gradient of the chemical potential and an external electric field. We analyse the similarities of the response arising from the vortex nodal-rings and the Hopf semimetals, which can be traced to the dipole-like sources in their BC fields.

Figures (4)

• Figure 1: (a) We show the unique features of a vortex nodal-ring (VRN) in terms of its dispersion against the $k_x k_y$-plane showing two bands crossing along a circle (highlighted in green), setting $k_z = 0$, and the profile of the vector fields representing the pseudospin and BC. (b) For the sake of comparison with a Weyl node, we show its dispersion (with the green dot denoting the nodal-point), and the pseudospin and BC distributions. The vector fields for each case have been drawn in the vicinity of a section of the Fermi surface for the positive-energy band.
• Figure 2: Profile of the BC-flux distribution for a monopole (e.g., Weyl node), an ideal dipole (e.g., three-band Hopf semimetal) [cf. Eq. \ref{['eqbcommhopf']} with $s=2$], and a Vortex nodal-ring [cf. Eq. \ref{['eqbcomm']} with $s=2$], projected on the $k_y k_z$-plane at $k_x =0$. While the white dot denotes the monopole singularity, the white arrows denote the dipole-like singularities of the BC.
• Figure 3: On introducing tilt, the Fermi surfaces take the forms of different kinds of cyclides, which depend on the values of $\mu$ and $\eta \, k_0$. The red and cyan colours depict the $s=1$ and $s=2$ bands, respectively. While (a) illustrates the full 2d Fermi surfaces, (b) shows the corresponding projections on the $k_x k_y$-plane for $k_z = 0$. The dotted black circle represents the original VNR at $\mu = \eta = 0$ for the sake of reference.
• Figure 4: Behaviour of $\delta \sigma_{zz}$ as a function of $B_z = \pm |\boldsymbol{B}|$ (in eV$^2$), setting $v_0 = 0.0004$ and $\beta_{\rm intr} = 1$. We have shown the curves for two different values of $\mu$, viz. $\mu = 0.01$ eV and $\mu = 0.04$ eV. While subfigure (a) represent the characteristics of a VNR, subfigure (b) depicts the response of a BC-dipole.